Long time dispersive estimates for perturbations of a kink solution of one dimensional cubic wave equations.

Abstract: A kink is a stationary solution to a cubic one dimensional wave equation $(\partial_t^2-\partial_x^2)\phi =\phi-\phi^3$ that has different limits when $x$ goes to $-\infty$ and $+\infty$, like $H(x) =\tanh(x/\sqrt{2})$. Asymptotic

stability of this solution under small odd perturbation in the energy space has been studied in a recent work of Kowalczyk, Martel and Mu\~noz. They have been able to show that the perturbation may be written as the sum $a(t)Y(x) +\psi(t,x)$, where $Y$ is a function in Schwartz space, $a(t)$ a function of time having some decay properties at

infinity, and $\psi(t,x)$ satisfies some local in space dispersive estimate.

The main result in this talk gives, for small odd perturbations of the kink that are

smooth enough and have some space decay, explicit rates of decay for $a(t)$ and for $\psi(t,x)$ in the whole space-time domain intersected by a strip $\abs{t}\leq \epsilon^{-4+c}$, for any $c>0$, where $\epsilon$ is the size of the initial

perturbation.

This is joint work with Nader Masmoudi.

analysis of PDEs

Audience: researchers in the topic

PDE seminar via Zoom